Abstract

The Open image in new window and Open image in new window problems are well-known problems in computer science. In this paper, we consider Open image in new window versions of both of these problems - Open image in new window and Open image in new window. For both problems, the input is a set of geometric objects \(\mathcal {O}\) and a set of points \(\mathcal {P}\) in the plane. In the MDIS problem, the objective is to find a maximum size subset \(\mathcal {O}^\prime \subseteq \mathcal {O}\) of objects such that no two objects in \(\mathcal {O}^\prime \) have a point in common from \(\mathcal {P}\). On the other hand, in the MDDS problem, the objective is to find a minimum size subset \(\mathcal {O}^\prime \subseteq \mathcal {O}\) such that for every object \(O \in \mathcal {O} \setminus \mathcal {O}^\prime \) there exists at least one object \(O^\prime \in \mathcal {O}^\prime \) such that \(O\,\cap \,O^\prime \) contains a point from \(\mathcal {P}\).

In this paper, we present \(\mathsf {PTAS}\)es based on Open image in new window technique for both MDIS and MDDS problems, where the objects are arbitrary radii disks and arbitrary side length axis-parallel squares. Further, we show that the MDDS problem is \(\mathsf {APX}\)-hard for axis-parallel rectangles, ellipses, axis-parallel strips, downward shadows of line segments, etc. in \(\mathbb {R}^2\) and for cubes and spheres in \(\mathbb {R}^3\). Finally, we prove that both MDIS and MDDS problems are \(\mathsf {NP}\)-hard for unit disks intersecting a horizontal line and for axis-parallel unit squares intersecting a straight line with slope \(-1\).

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S. Pandit—The author is partially supported by the Indo-US Science & Technology Forum (IUSSTF) under the SERB Indo-US Postdoctoral Fellowship scheme with grant number 2017/94, Department of Science and Technology, Government of India.